I have two sets $A$ and $B$ which are subsets of a Hilbert space, which are both cones (i.e. $a \in A$ implies that $\gamma a \in A$ for all $\gamma > 0$, and likewise with $B$). Furthermore, $B$ is a closed set.
Is there a nice formula for $\overline{A \cap B}$ (the closure) in terms of $\bar A$ and $B$? I know there is always
$\overline{A\cap B} \subset \bar A \cap B$, but what about a reverse?
There is no formula for $\overline{A\cap B}$ in terms of $\overline{A}$ and $B$. For instance, let $A$ be any non-closed cone containing $0$, let $b\in \overline{A}\setminus A$, and let $B=\{cb:c\in[0,\infty)\}$. Then $\overline{A\cap B}=\{0\}$. But, if you take $A'=\overline{A}$ then $A'$ has the same closure as $A$ and $\overline{A'\cap B}$ contains $b$ and so is not the same as $\overline{A\cap B}$. So, $\overline{A\cap B}$ is not determined by $\overline{A}$ and $B$.